Robust Intermediate Positions in a Permutation

Let $\sigma$ be a uniformly random permutation of $\{1, 2, \dots, n\}$. Call position $i \in \{2, \dots, n-1\}$ an intermediate position if $\sigma(i)$ is strictly between $\sigma(i-1)$ and $\sigma(i+1)$, i.e., $\min(\sigma(i-1), \sigma(i+1)) < \sigma(i) < \max(\sigma(i-1), \sigma(i+1))$. What is the expected number of intermediate positions?

Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.